Stochastic Volatility Modeling

232 Stochastic volatility modeling

120%

130%

140%

150%

160%

170%

180%

50% 100% 150% 200%

Set I

Set III

Figure 7.4

: Implied volatilities of variance swaptions – that is of

bσ

2

T

1

T

2

(T

1

)

– as a

function of

volatility

moneyness:

√

K/bσ

T

1

T

2

(t = 0)

with

T

1

= 3

months,

T

2

= 6

months in sets I and III. Dotted lines correspond to the strike-independent level

2bν

T

1

T

2

(T

1

), where the integral in (7.41) has been computed numerically.

(7.26) over [0, T

1

].

2bν

T

1

T

2

(T

1

) = 2

s

1

T

1

Z

T

1

0

ν

2

T

1

T

2

(t) dt (7.41)

2bν

T

1

T

2

(T

1

) appears in Figure 7.4 as a dashed line.

Figure 7.4 displays the weak positively sloping smile that is typical of baskets

of lognormal underlyings: while not exactly lognormal, VS ...